Oaksford, Michael and Chater, N. (2016) Probabilities, causation, and logic
programming in conditional reasoning: reply to Stenning and Van Lambalgen
(2016). Thinking & Reasoning 22 (3), pp. 336-354. ISSN 1354-6783.
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Probabilities, Causation, and Logic Programming in
Conditional Reasoning: Reply to Stenning and Van
Lambalgen
Mike Oaksford*
Birkbeck College, University of London
Nick Chater
Warwick Business School, University of Warwick
Short Title: Probabilities, Causation, and Logic Programming in Conditional Reasoning
*Corresponding author:
Mike Oaksford
Department of Psychological Sciences
Birkbeck College
University of London
London, WC1E 7HX, UK
e-mail: [email protected]
Abstract
Oaksford and Chater (2014) critiqued the logic programming (LP) approach to
nonmonotonicity and proposed that a Bayesian probabilistic approach to conditional
reasoning provided a more empirically adequate theory. The current paper is a reply to
Stenning and van Lambalgen’s (in press) rejoinder to this earlier paper. It is argued that
causation is basic in human cognition and that explaining how abnormality lists are created in
LP requires causal models. Each specific rejoinder to Oaksford and Chater’s (2014) original
critique is then addressed. While many areas of agreement are identified, with respect to the
key differences, it is concluded the current evidence favours the Bayesian approach, at least
In their rejoinder to our paper in the recent special issue on dual processes in this journal,
Stenning and Van Lambalgen (2015, henceforth “SL”) argue that we have misrepresented
their logic programming approach (henceforth, “LP”) to human reasoning. We should say at
the outset that we are grateful to SL for the constructive nature of their response and that we
hope that our reply will be interpreted in a similarly constructive vein. Following SL, we
focus our response on our argument that causal Bayes nets (CBNs) may provide a useful
formalism for theorising about conditional inference (Ali, Chater, & Oaksford, 2011; Ali,
Schlottmann, Shaw, Chater, & Oaksford, 2010; Fernbach & Erb, 2013; Chater & Oaksford,
2006; Oaksford & Chater, 2010a, 2013, 2014, in press). However, the application of
probability theory to the psychology of human reasoning goes beyond this recent proposal.
There is currently a range of probabilistic approaches, which go under the rubric of The New Paradigm (Manktelow, 2012; Over, 2009), which have many commonalities but some important differences (e.g., Baratgin, Over, & Politzer, 2013; Douven & Verbrugge, 2013;
Fugard, Pfeifer, Mayerhofer, & Kleiter, 2011; Gilio & Over, 2012; Over, Hadjichristidis,
Evans, Handley, & Sloman, 2007; Politzer, 2005; Pfeiffer & Kleiter, 2010). In particular, our
approach has been Bayesian from the outset (Oaksford & Chater 1994) but has also been
influenced by Adams (1998) work on probability logic and work on CBNs (Pearl, 1988;
2000). In this reply, while focusing on CBNs, we will appeal to a range of probabilistic
approaches taken in the literature. This is because it is clear from SL’s response that the use
of probabilities in theorising about human reasoning and language interpretation is the
primary target of their rejoinder.
Some Intuitions
We first outline two intuitions that guide our approach to the psychology of conditional
It’s causal, all the way down.
The origins of the main contemporary accounts ofthe conditional are based on intuitions
about causal or law-like dependencies. In the philosophy of science, it was soon realised that
the material conditional (p⊃q), which is true as long as p is false or q is true, could not account for such dependencies (Chisolm, 1946; Goodman, 1947, 1955). Law-like relations
support counterfactuals. That is, the reason we believe that the counterfactual (1),
If the key had been turned the car would have started (1) to be true is that we believe that the indicative (2)
If the key is turned the car starts (2)
describes a real causal dependency in the world (Stalnaker, 1984). Notice that according to
the material conditional, counterfactuals are always true, because their antecedents, i.e., p, are always false. This is, after all, why they are counterfactuals. Consequently, the material conditional could not capture our intuitions about counterfactuals.
The possible worlds semantics for the counterfactual (Lewis, 1973; Stalnaker, 1968)
is widely recognised as the beginning of our understanding of the conditional of natural
language as opposed to in mathematics (Nute, 1984). Agreeing with SL on the importance of
the Ramsey test, both probability logic (Adams, 1998) and the possible worlds semantics for
conditionals (Stalnaker, 1968, 1984) have their origins in this test. For example, the truth of
(1) involves considering the possible world most similar to the real world in which the key
was turned, if the car starts in this world then the conditional is true. Exactly the same
formulation applies to “open” indicative conditionals where it is not known whether the
antecedent is true or false (2). Stalnaker’s (1984) conceptualist interpretation of possible
worlds also indicates how causality is fundamental to our evaluation of conditionals. On this
interpretation, open indicative conditionals correspond to our methodological policies for
methodological policy in (2) describes a real causal dependency in the world. This is a claim
about the person’s psychology: we can remain neutral on the ontological question of whether
such a relation really exists. All that matters is that our methodological policies, or as Hume
referred to them our habits of inference, are believed to denote relations in the world that
behave like causal relations. This is why we have argued that these habits of inference and
hence the conditionals used to describe them can be represented in the structure of causal
Bayes nets (Ali, Chater, & Oaksford, 2011; Chater & Oaksford, 2006; Oaksford & Chater,
2013). From this point of view, the way people think about the world is causal “all the way
down” and causality is not reducible to anything else, like probabilities or similarity between
possible worlds (Oaksford & Chater, 2010a, in press) .1
Some caveats are in order. Causal Bayes nets and the probabilistic inferences they
allow are useful because there is uncertainty about the nature and efficacy of the causal
relations that are operative in any given situation. The structure of a network relies on our
causal intuitions but it and the strength of causal relations can be learned from data such as
the direct observation of occurrences of putative causes and effects (Griffiths & Tenebaum,
2005), although this relies on some strong assumptions (i.e., faithfulness). But the input
people receive may also consist of testimony from other people in which a causal or law-like
relations are described using conditionals.
The received view of dual processes.
While there may be some room for disagreement, given thelargely informal discussions of
dual processes, there is a generally received view of System 1 as associative and heuristic and
System 2 as being rule bound and tied to language. One might initially question how
associative and heuristic go together but the availability heuristic provides a clear example.
This heuristic emerges as the result of world knowledge in LTM being represented in
associative or connectionist networks in which the speed of retrieval is related to the strength
of connections. Speed or ease of retrieval then provides a heuristic for judged probability.
Given the body of evidence cited to support the basis of this division between these two
systems (see, Evans, 2009; Sloman, 1995; Stanovich, 2011), a broad division of this kind
seems a reasonable starting point for discussion, although many of the details may be
contested. Indeed, a recent paper by Sternberg and McClelland (2012) provides relevant
evidence for this type of division in a context closely related to the present analysis. Using a
human contingency learning paradigm, they found that, under speeded conditions, their
results were consistent with a pathway strengthening process like that implemented in
connectionist or associative networks. However, with no time constraints, performance was
consistent with an inferential approach based on choosing between possible causal models.
Response to Stenning and Van Lambalgen
As we just pointed out, the fundamental intuition about our use of CBNs to represent
conditional statements was the relation to a non-reductive sense of causality. In this regard,
LP and our approach are formally similar. This is because LP has been used to implement the
Event Calculus (Kowalski & Sergot, 1986) and under certain conditions, “the event calculus
would correspond to directed acyclic graphs that are equivalent to Pearl's causal structures”
(Sowa, 2000/2015). In the event calculus, causation is also treated non-reductively. Bayesian
networks and the event calculus provide alternative representations of causal relations, in just
the same way that syntactic mental logics and mental models provide alternative mental
representations of logical relations. Our contention is that the empirical evidence supports the
approach. This is borne out in the first section of SL’s rejoinder, which, after a brief
explication of the LP approach, looks at applications of LP in the empirical literature.
Applications of LP.
The list of applications of LP to empirical phenomena is not long. This contrasts with the
Bayesian cognitive science framework, which we and others apply to human reasoning. The
empirical reach of this framework is evidenced by edited collections spanning the last 20
years (e.g., Chater & Oaksford, 2008; Oaksford & Chater, 1998).
In this section, SL also provide an example of how LP accounts for suppression
effects. They then describe some empirical findings (Pijnacker et al, 2009, 2010) that
replicate the suppression effect for the MP inference. We first critique SL’s example as it is
provides a confusing illustration of how LP works which needs clarification. The example
inference is as follows (SL, p. ??):
“MP-Disabling
Lisa probably lost a contact lens. L (disabling condition)
If Lisa is going to play hockey, she will wear contact lenses. H ∧ ¬ab → (conditional) L →ab (hidden premise from abnormality list expressing that L is disabling)
Lisa is going to play hockey. H (categorical premise) Lisa will wear contact lenses. W (conclusion)
MP-Neutral
The formalisation is exactly the same except that there is no hidden premise L →ab, because buying contact lenses is not on the ab list.”
This example is confusing because the disabling condition and the consequent of the
conditional/conclusion (W) share content. SL describe L (probably losing a contact lens) as disabling for playing hockey (H). But regardless of whether Lisa plays hockey, L is not consistent with the conclusion W,that she will wear contact lenses. Probably losing a contact lens is as disabling for wearing contact lenses as it is for playing hockey. Indeed, people may
one might hunt for one’s contact lenses, without realizing one is wearing them, but the
possibility seems remote). Consequently, the conclusion seems to be contradicted without the
need to contemplate whether Lisa is playing hockey or not. That is, non-monotonic reasoning
need have nothing to do with this inference.
Moreover, SL describe the MP-Neutral condition as simply missing the hidden
premise L → ab. By being “hidden” we assume this premise was not mentioned in the task and that it is just part of the computation in LP. But this raises the question: what is the difference in the materials presented to participants between the MP-Neutral2 and the MP-Disabling conditions? Normally, using the explicit Byrne (1989) paradigm, L would not be presented in the MP-Neutral condition but the text seems to suggest that it was—apparently only the hidden premise L → ab was omitted in the computation. If L was presented in the
MP-Neutral condition, then psychologically it seems odd that people could not infer from world knowledge that L is disabling for Lisa playing hockey, whether it was on an
abnormality list for the conditional rule or not. Moreover, if it was not, then why is explicit
mention of L required to put it on the abnormality list? For this conditional, where the
consequent W shares content with the disabling condition, considering the possibility that she loses her contact lens yields the conclusion, she will not play hockey, by MT. That is, it is
clear that losing a contact lens is disabling for Lisa playing hockey. The problems with this
example show the care that needs to be taken in setting up genuine instances of defeasible
reasoning. We suggest that the original example we borrowed from Wernicke better
exemplifies how LP handles suppression effects.3
LP and Dual Process Theories.
2 We do not understand the purpose of the reference to “buying contact lenses” in the description of the
MP-neutral condition.
In this section, SL first express their bafflement at our proposal that Stenning and van
Lambalgen (2005) gives the impression that LP holds out the promise that non-monotonic
reasoning can be handled by purely local System 2 computations. We are equally puzzled:
the example of how LP explains the suppression effect in our original paper (Oaksford &
Chater, 2014, p. 275, programs A and B) was taken verbatim from Stenning and van
Lambalgen (2005, pp. 942-943). These programs require only a few local steps in order to
draw suppression inferences. However, SL suggest that only “a cursory reading” of the
literature on LP is required to encounter the publication SL cite (i.e., van Lambalgen &
Hamm, 2004), 4 on how knowledge importation, i.e., System 1, works in LP. However,
Stenning and van Lambalgen (2005) remains the primary source for the relevance of LP to
psychological phenomenon such as suppression effects and we stand by our comments as an
important corrective to the mistaken impression to which the discussion in the 2005 paper can
easily give rise. Moreover, in our original paper we also stressed that LP can handle the
computational complexities of knowledge importation. What we questioned was the
psychological reality of the LP approach. This is the criticism to which SL turn next.
SL quote us saying, “…, how plausible is it to propose that people are explicitly
indexing exceptions as abnormal.” Their counterargument is to suggest that what Cummins
(1995) refers to as “disablers” are just abnormalities and thus the effects observed are
therefore, by definition, evidence for the psychological reality of abnormalities. But this is
too quick. Our argument was against the psychological reality of abnormality propositions or
predicates, ab, indexing a proposition or a subject as abnormal with respect to a rule. In a realistic data base, there have to be large numbers of different ab propositions/predicates, one for each rule. These abnormality propositions/predicates are a part of the deep logical
structure underlying the interpretations of almost all conditionals. These are the LP
representations of our conditional beliefs. Yet, as we argued in our original paper, these
abnormality propositions/predicates leave no trace in the surface linguistic forms or in our
natural way of speaking about these beliefs.5
As we indicated in the opening section, we think causality comes first. We can see
why by considering the role of the abnormality list and how items get onto this list.
According to SL, the MP-Neutral case just needs to miss out the L → ab premise for MP to go through. Let us assume that for a particular reasoner L is not on their abnormality list, that is, the L → ab premise is missing. If L is then presented along with the conditional and categorical premises, how does L get on to the participant’s abnormality list, as common sense seems to indicate it should? The way to do this is to build a causal model of the
situation and simulate whether the presence of L prevents Lisa wearing contact lenses. In SL’s example, where L and W share contents—they are both about contacts lenses—this is straightforward but it is less so for more realistic examples.
The need to construct causal models also derives from another part of the procedure
used in experiments like Cummins (1995), where in a pre-test participants are asked to
generate as many disablers or alternative causes as they can in one minute. For LP,
presumably, this just involves reading the items off the abnormality list for the rule. But this
could not be the case for the novel rules used in these experiments. For these, there are no
precompiled rules with an associated abnormality lists. On our view, people build causal
models of the situation described in the conditional, recruit semantic associates of those
materials and check whether in their causal model the presence of the putative disabler
prevents the consequent event occurring.
SL also cite evidence from autistic participants (Pijnacker, et al. 2009, 2010) for the
psychological reality of abnormalities or exceptions. Of course, it is not the bare reality of
exceptions against which we argued, but LP’s account of how they are handled. The
experiments they cite would be more compelling if LP were the only theory that predicts
apparently dysfunctional exception handling and these were the only experiments
demonstrating this phenomenon. However, we have shown dysfunctional exception handling
in high schyzotypes (Sellen, Oaksford, & Gray, 2005). High schizotypes showed no effect of
many disablers in a replication of Cummins (1995) with high and low schizotypy groups.
That is, they did not show a suppression effect for MP or MT but they did show a suppression
effect for AC and DA. The negative symptoms of schizophrenia are thought to be due to
reduced dopamine levels in the frontal lobes. Cohen and Servan-Schreiber (1992) modelled
some of the observed cognitive symptoms by turning down the gain parameter of
connectionist units representing context. We modelled our findings with high schizotypes in a
similar way (Oaksford & Chater, 2010b). The absence of exceptions provides the context in
which a rule will work or a cause will produce its effect. Consequently, in a neural network
representation of System 1, we turned down the gain of the units representing disablers,
reducing their effect on inference when they were included in the premises. This one
parameter change provided good fits to Sellen et al.’s (2005) results. Interestingly, this may
link with a relatively old finding that dopamine levels are reduced in autism (Lake, Ziegler, &
Murphy, 1977).
In this section, SL also describe our citing hash coding as an analogy for LP’s
abnormality mechanism as “egregiously misleading.” They argue that this is because, “It is
no good reacting against all immigration of formalisms from outside psychology: one simply
loses the benefits of mathematics.” Of course, we never argued against all immigration of
formalisms from outside psychology, just LP. Indeed, we have specifically argued for the importation of the causal Bayes net formalism. Like SL, we fully support the need for
the hash coding analogy was apposite: both hash coding and abnormality propositions solve a
computational problem in a way that seems to assume answers to the psychologically
interesting questions. In the case of LP, the psychologically interesting questions are largely
epistemic—e.g., how do we construct an abnormality list? Specifically, we doubt that LP
solves the epistemological frame problem (Fodor, 1983, 2001; Shanahan, 2009).
In the closing paragraph of this section, SL discuss their view of the relationship
between System 1 and System 2. We are in agreement with much of what they have to say
here. We (Oaksford & Chater, 2013; in press) have also argued that people construct small
scale models of the premises of arguments (integrated with world knowledge) from which
they “read off” putative conclusions.6 The details are beyond of the scope of this reply but we
can highlight the contrasts with SL. We envisage these models as causal models where
associative knowledge (represented in connectionist networks) may be used elaboratively to
suggest further structure (in addition to that conveyed by the premises) and the values (or
distributions of values) of the various parameters. Often, in order to make a response, more
than one interrogation of these models is required (whether alternative models are considered
or not). We have argued that the need to verbalise the results of the records of these
interrogations may lead to systematic errors (Oaksford & Chater, in press). But these errors
would be at the System 2 level. The contrast with SL is clear, where we envisage System 1 as
associative and connectionist, they view it as provided by a symbolic LP data base.
Moreover, where we envisage the small scale models as similar to CBNs, they view them as
provided by localist7 neural networks. The inversion of System 1 process proposed by SL in
6 Of course, this is hardly a novel proposal (e.g., Johnson-Laird, 1983) although there is now active debate about the nature of these small scale models of the world which people construct in order to draw inferences.
what we called in the introduction “the received view” of the System1/System 2 distinction
could not be clearer.8
Replies to Specific Criticisms.
In the main section of the paper, SL respond to what they take to be our principle criticisms
of LP. We address each of their responses in turn (SL’s numbering is shown in brackets).
Inconsistency (1). In this section, SL query our specific astronomical example for
being “exceptional” and then, after an observation about the representation of “theoretical
and observational conditionals,”9 they outline a variety of ways in which LP may handle
inconsistencies. They note, anecdotally, that there are similarities with human experience.
While this is an interesting discussion, it sidesteps the fact that we discussed the response to
inconsistency in three different sections of our paper.
In one of these sections, we discussed how perceived inconsistencies can be treated as
an occasion to learn, i.e., to revise the strength of a perceived dependency between antecedent
and consequent. Moreover, we have shown how this strategy can provide much better fits to
the experimental data on abstract conditional inference tasks (Oaksford & Chater, 2007;
2013). These data consisted of 65 individual studies using the conditional inference task
involving 2774 participants. That is, this approach to inconsistency, which is not available to
LP, in which there is no direct analogue of altering strength parameters,10 provide good fits to
a large amount of data. This is in contrast to LP, where there is no more than anecdotal
support for its mechanisms for handling inconsistency and for which no detailed model fits
have, to our knowledge, so far been provided.
8 We also agree with SL that a lot of the action is at the System 1/model interface and that System 2, involving language and handling alternatives, remains very underspecified in dual systems approaches.
9 This comment was rather obscure and we would value seeing it elaborated.
10 In the noisy–OR rule, the strength of a dependency is treated as inversely proportional to the number of disablers, which is partially consistent with the approach SL discuss later. However, in the noisy–OR
SL (p. 5) further suggest that the Bayesian probabilistic approach “does not account
for inconsistency handling as much as blur it.” Yet there is good evidence that people are a
sensitive to this “blurred” notion (Cruz, Baratgin, Oaksford, & Over, 2015). According to a
probabilistic account based on what Edgington (1995) called the Equation, that is, Pr(if p then q) = Pr(q|p), the probabilities of the premises of an inference place coherence constraints on the range of values that the probability of the conclusion can take. Moreover, these coherent
probability intervals for the conclusion are a deductive consequence of the premises and their
probabilistic interpretation. It has been shown that peoples’ assignments of conclusion
probabilities respect probabilistic coherence (Cruz et al., 2015). This is not the same as
showing that people are sensitive to violations of coherence but at least one possible reaction
to detecting such a violation is clear from our learning example. They need to revise one of
their premise probabilities. This is an interesting area for future investigation; indeed, we are
experimentally investigating people’s reaction to coherence violations currently.
Relevance (2). We suggested that in addressing the epistemological frame problem
LP would need to address the computation of relevance in System 1. We mentioned also that
context occasionally helps to disambiguate relevant interpretations. SL respond by arguing
that LP has been directly applied to computing contextual relevance. On this point, we would
concede that LP may well be making some theoretical advances. Our own view on these
issues is that mechanisms like online schema computation in constraint satisfaction neural
networks (Rumelhart, Smolensky, McClelland, & Hinton, 1988) probably underpin the
computation of relevance. Moreover, in the papers SL cite elsewhere (e.g., Baggio, Choma,
van Lambalgen, & Hagoort, 2010, p. 2137) discussing the ERP evidence on how LP
computes contextual relevance, it is conceded that, “…the ERP data reported here do not
favour unification [the LP approach] over alternative accounts…” (our italics). So the
satisfaction network, when there is no disambiguating contextual information available, the
strength of connections will make different interpretations more available than others. As we
indicated above, we doubt that the solution LP provides to the narrow, technical frame
problem can account for the broader epistemological frame problem. This would require an
account of analogy and metaphor (Shanahan, 2009) which has been argued to be the core of
the unconscious reasoning system (Lakoff & Johnson, 1980, 1999). In this respect,
connectionist models of System 1 may be in better shape (e.g., Thomas & Mareschal, 2001).
Enablers vs. alternative causes (3). Distinguishing the status of premises in the
suppression task seems not to yield any real disagreement between us and SL. SL concede
that these distinctions require general knowledge. However, the proposal that enablers will be
represented as conjunctive antecedents (p ˄ s ˄ ab' → q) does not seem to distinguish causes (p) from enablers (s). In contrast, there is a detailed causal model theory of these distinctions (Sloman, Barbey, & Hotaling, 2009). That is, we believe that this distinction is not “one that is as problematic, or not, in BNs as LP nets.”
The implicit suppression paradigm (4). In this section, SL respond to our
suggestion that the implicit suppression paradigm (e.g. Cummins, 1995) is problematic for
LP. They argue that this paradigm shows that an alternative, more qualitative, LP approach
to suppression effects is possible. This initial argument presupposes that there is a good
correlation between the magnitude of suppression effects and the number of defeaters or
alternative causes participants can generate in a pre-test. The pre-test in the implicit paradigm
is usually carried out with a separate group of participants and SL do not cite any literature on
the studies demonstrating the extent of this correlation. In, for example, Geiger and Oberauer
(2007), number of disablers was an independent variable in a factorial design.11 We do not
doubt that the correlations SL mention will be observed but the upshot of Geiger and
11 In the similar factorial designs used by Cummins (1995), effect sizes were not reported so we cannot
Oberauer’s studies was that information about the number of different disablers is completely
overridden when frequency information is available.
SL go on to propose that frequency information can be generated from the operation
of LP—that is, they suggest that such information can be gleaned from intensional
knowledge. We agree that this is a potentially important development. However, there are
two points to make. First, the proposed mechanism apparently involves a neuron in the neural
network implementation counting the times a rule/abnormality proposition is activated. This
is an ad hoc addition to the proposed implementation of LP. Since it is not inherent to the LP
approach it could just as well be implemented in a similar ad hoc manner in other approaches
if required. Second, SL provide no reference for this work so that it can be judged
independently, so it is not clear how far this approach has been developed. Nonetheless, we
assume that the discussion here is intended to feed forward to SL’s discussion of Martignon
et al. (in prep).
Before discussing some further aspects of Martignon et al. (in prep), SL diverge in to
two long quotations from Pearl about the need for additional sources of information in order
to extract causal information from raw data. This was in response to our argument concerning
abnormality lists that what is normal or abnormal must be learnt from the statistical structure
of the world. As we pointed out in the first section of this paper, we have argued that
causation is basic. Our collective intuitions about causation from many sources allow us to
form hypotheses about the structure of the world which statistical evidence can confirm or
deny. This knowledge allows us to form causal hypotheses captured in the structure of a
causal Bayes net. However, a great deal of what we are told we just take to be true without
question, if it coheres with what we already believe. Bayesian accounts provide a formal
measure of coherence (Bovens & Hartmann, 2003) for which there is some experimental
abnormal has to be learned; and it is learned by discovering, by whatever means, the causal structure of our world. As Sternberg and McClelland (2010) have recently showed, we can
learn the nature and efficacy of causes either inferentially or by pathway strengthening or, of course, we can just accept the testimony of a reliable informant (Hahn, Harris, & Corner,
2009; Hahn, Oaksford, & Harris, 2013; Oaksford & Hahn, 2013).
In the final paragraphs of this section, SL mention the experimental work (Martignon
et al., in prep) which underpins their earlier comments. They report experiments using a
within subjects design. They found that ratings of the likelihood of disablers explains
additional variance in predicting the inferences participants endorse, e.g., MP, over and above
the sheer number of disablers generated. This experiment appears to replicate Fernbach and
Erb (2013) where participants estimated disabling strength and the base rate of disablers to
compute causal power in a noisy-OR gate. For each retrieved disabler, the disabling
probability can then be calculated as disabling strength × base rate. Causal power (Cheng,
1997) is equal to 1 minus the aggregate of the disabling probabilities (see, Ferbach & Erb,
2013, Eqs. 3, 4, & 5; Oaksford & Chater, in press, Eqs. 8 & 9). So causal power as a predictor
of endorsements of MP or MT captures both the number of disablers and what SL refer to as
likelihood. Consequently, the data SL describe would seem unable distinguish between their
ad hoc extension of LP and causal Bayes nets. Both seem to predict that number of disablers
and disabling probability (“likelihood”) will affect MP and MT ratings or endorsements.
Explaining away alternative causes (5). In this section, SL suggest that LP can
explain discounting inferences in contradistinction to the argument in our original paper.
However, they have misrepresented these inferences. Using their formal example,
discounting occurs when comparing two situations (“r?” is the question posed to participants in the task):
p → q, r→ q, q, p, r? (4)
For example, where p: it is raining, r: the sprinklers are on, and q: the pavements are wet, learning that that the pavements are wet (1) makes it quite likely that the sprinklers are on (r). So, when asked “are the sprinklers on?” (r?), participants should respond that this is highly probable. However, finding out that it is raining discounts the sprinklers being on as the explanation of the pavements being wet—the rain alone explains why the pavements are wet,
so that the mere wetness of the pavements provides no indication of whether the sprinklers
were on or not. In the experimental data, both children (Ali, et al., 2010) and adults (Ali, et
al., 2011) rated r as less likely in (4) than in (3). But SL observe that in LP for both (3) and (4), r is not derivable. Consequently, people should not endorse r given either set of premises so there should be no differences between conditions. The case SL consider is not the actual
discounting inference involving the contrast between (3) and (4) but the contrast between (3’)
and (4):
r→ q, q, r? (3’)
In (3’) r is derivable, in a way that it is not in (3). All the experimental tasks in Ali et al (2010, 2011) used the premise sets in (3) and (4). Consequently, SL fail to show how LP can
account for the discounting inferences observed in these experiments. They suggest the
possibility that in (3) the disjunction p ˅ r might be derivable at the System 2 level, that is,
outside of LP. But to see a reduction in endorsements in (4) would seem to require an inference of the form p ˅ r, p therefore not-r, which would require the further ad hoc assumption that “˅”is interpreted exclusively.
(3) and (4) are the context in which the discounting inference has been described in
the normative literature. Discounting follows as a consequence of the Markov condition in
causal Bayes nets; without a similarly strong structural assumption it is not clear how LP can
largely been explained within the causal Bayes net framework (Rehder, 2014). In sum,
discounting is indeed a novel prediction of the Bayesian approach (Chater et al., 2011) one
which we suggest that SL cannot explain within LP—unspecified System 2 inferential mechanisms must be invoked.
Conclusion
In concluding, we first observe that SL do not address the closing sections of our paper. We
then raise some issues about where this debate could go in the future. Finally, we consider the
where the burden of proof lies in this debate.
Our speculative closing remarks were aimed at providing an answer to the practical
question of when in real life do we need consistent,12 retrievable, default hierarchies? We
suggested that (1) consistency is probably a local issue about the particular models we build
to guide our actions in the world and not to be expected of our global world knowledge; and
that (2) knowledge about most situations is shallow which is remediated by the fact that
knowledge is socially distributed. We could add here that as we (Hahn & Oaksford, 2007)
and others (Mercier & Sperber, 2011) have argued reasoning is engaged primarily in the
service of argumentation, that is, in resolving disagreements between people. Thus, when we
make public commitments we are often required to argue for their consistency with our other
beliefs or the facts. Making a public commitment could also be to act on our beliefs, with
inconsistency becoming apparent if those actions fail. It is only in these behavioural and
social settings that consistency becomes a serious issue. Consequently, our individual global
belief systems may not need to satisfy such exacting standards as global consistency. This is a
potentially deflationary approach to the problem that LP tries to solve and perhaps to the
frame problem in general—that is, in psychologically real belief systems this problem doesn’t
actually exist.13 From the web of soft constraints that makes up our associative world
knowledge we construct small scale models that provide the content of our thoughts and the
basis for our actions the contents of which we can verbalise in language and communicate to
others; but it is only at this final stage that consistency and resolving consistency is an issue
and it is a local issue about a particular model.
In furthering the debate between LP and the probabilistic approach based on causal
Bayes nets or probability logic there are a couple of issues worth mentioning. First, both
suggest a three valued system, for LP it is Kleene’s strong three valued system (Haack, 1974;
Stenning & van Lambalgen, 2005) and for probability logic it is de Finetti’s three valued
system (Baratgin, Over, & Politzer, 2013). These systems may differ on their interpretation of
the third truth value (Haack, 1974). However, the actual truth tables are identical (Baratgin et
al., 2013). This means that at this level of abstraction from continuous valued probabilities
(which SL don’t like) there may be no grounds to distinguish LP from probabilistic
approaches. It is only when we look at probabilistic inferences, like discounting, that
differences emerge. . Second, at a deep theoretical level there is a close connection between
Bayes nets and LP (Kowalski, 2010; Poole, 1993, p. 81)—“any probabilistic knowledge
representable in a discrete Bayesian belief network can be represented in…a simple
framework for clause abduction, with probabilities associated with hypotheses.”
Horn-clause abduction is the inferential scheme employed in LP. Working out these relationships
may provide for some further productive dialogue.
In their closing paragraph, SL (p.??) place the burden of proof, “on those who would
invoke probability…to show that it can do something that crude intensionally derived
conditional frequencies of inference cannot.” The short response to this suggestion is
“explain some of the empirical data.” The paper reporting this new mechanism has not been
published, so it cannot yet be judged whether it can be applied to the range of experimental
results to which probabilistic approaches have been applied. So, the burden of proof lies with
LP to demonstrate that it is capable of explaining the large range novel results spawned by
probabilistic approaches.14
In conclusion, we thank SL for their constructive response to our critique of LP. We
think many issues have been clarified. On the issue of dual systems there are points of
agreement—the System 1/model interface and the current underspecification of System 2—
but also important areas of disagreement—we cleave much closer to the received view of the
System 1/System 2 distinction. The fundamental disagreements surround the treatment of
nonmonotonicity and uncertainty and whether this only a local or global problem for real
human reasoners. As with the comparison between all theories, eventually the data will be the
ultimate determiner of which approach is correct and in this respect we suggest that
probabilistic approaches seem to have the edge, at least for the moment.
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